cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A109574 Chen primes p such that p is palindromic.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 181, 191, 353, 787, 797, 919, 10301, 10601, 11411, 12721, 12821, 13331, 13931, 14741, 15551, 16061, 16361, 16561, 16661, 17471, 19991, 31013, 35753, 36263, 38783, 71317, 72227, 73037, 73237, 73637, 74047, 74747
Offset: 1

Views

Author

Jason Earls, Aug 30 2005

Keywords

Links

Crossrefs

Intersection of A002113 and A109611.

Programs

  • Mathematica
    chenQ[n_] := PrimeQ[n] && PrimeOmega[n + 2] < 3; Select[Range[75000], chenQ[#] && PalindromeQ[#] &] (* Amiram Eldar, Oct 19 2021 *)
  • PARI
    isok(p) = my(d=digits(p)); isprime(p) && (bigomega(p+2) <= 2) && (d==Vecrev(d)); \\ Michel Marcus, Oct 19 2021

A118482 Partial sums of Chen primes (starting with 1).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 59, 78, 101, 130, 161, 198, 239, 286, 339, 398, 465, 536, 619, 708, 809, 916, 1025, 1138, 1265, 1396, 1533, 1672, 1821, 1978, 2145, 2324, 2505, 2696, 2893, 3092, 3303, 3530, 3763, 4002, 4253, 4510, 4773, 5042, 5323, 5616, 5923
Offset: 0

Views

Author

Jani Melik, May 05 2006

Keywords

Links

Crossrefs

Cf. A014284, A109611.

Programs

  • Maple
    ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_partsum_chenprime:=proc(n) local i,ans,tren; ans:=1: tren:=1: for i from 1 to n do if (ischenprime(i)='true') then tren := tren+i: ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_partsum_chenprime(500);
  • Mathematica
    Accumulate[Join[{1},Select[Prime[Range[70]],PrimeOmega[#+2]<3&]]] (* Harvey P. Dale, May 26 2014 *)

A269256 Chen primes p such that there are Chen primes p > q > r in arithmetic progression.

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 113, 127, 131, 137, 139, 149, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 347
Offset: 1

Views

Author

Charles R Greathouse IV, Jul 12 2016

Keywords

Comments

Green & Tao prove that this sequence is infinite.

Examples

			19 is in the sequence since 3 < 11 < 19, 19 - 11 = 11 - 3, all three are prime, and 3+2, 11+2, and 19+2 are each either prime or semiprime.

Links

Crossrefs

Subsequence of A109611. This is the Chen prime analog of A216495.
Cf. A291525.

Programs

  • PARI
    issemi(n)=bigomega(n)==2
ischen(n)=isprime(n) && (isprime(n+2) || issemi(n+2))
is(n)=if(!ischen(n), return(0)); forprime(p=2,n-4, if((p+n)%4==2 && ischen(p) && ischen((p+n)/2), return(1))); 0

A280009 Orders of consecutive clusters of Chen primes.

Original entry on oeis.org

13, 3, 2, 2, 1, 8, 1, 1, 3, 3, 1, 2, 4, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 8, 1, 6, 1, 1, 2, 3, 2, 1, 1, 2, 2, 5, 3, 1, 3, 1, 2, 4, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 5, 1, 3, 3, 1, 1
Offset: 1

Views

Author

Enrique Navarrete, Feb 21 2017

Keywords

Links

Crossrefs

Cf. A109611.

Programs

  • Mathematica
    Length /@ DeleteCases[Split@ Table[Boole@ And[# != 1, PrimeOmega@ # <= 2] &[# + 2] &@ Prime@ n, {n, 300}], k_ /; Total@ k == 0] (* Michael De Vlieger, Feb 27 2017 *)
  • PARI
    do(lim)=my(v=List(),u=v,r,s=1); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), listput(u,p*q))); u=Set(u); r=3; forprime(p=5,lim, if(p-r==2 || setsearch(u, r+2), s++, if(s, listput(v, s); s=0)); r=p); u=0; Vec(v) \\ Charles R Greathouse IV, Feb 27 2017

A109504 Chen primes p such that p + 2 is triangular.

Original entry on oeis.org

13, 19, 53, 89, 251, 701, 1709, 1889, 2699, 5669, 12401, 13859, 18719, 38501, 49139, 60029, 104651, 114479, 146609, 158201, 188189, 226799, 258119, 371951, 385001, 497501, 597869, 665279, 954269, 1034639, 1159001, 1309769, 1660751, 1869209
Offset: 1

Views

Author

Jason Earls, Aug 29 2005

Keywords

Examples

			a(9) = 2699 because it is prime and 2701 = 37*73 and 73*74/2 = 2701.

Links

Crossrefs

Cf. A000217, A109611.

Programs

  • Mathematica
    tri[n_] := n*(n + 1)/2; Select[tri /@ Range[2000] - 2, PrimeQ[#] && PrimeOmega[# + 2] == 2 &] (* Amiram Eldar, Dec 17 2019 *)
Select[Prime[Range[150000]],(PrimeQ[#+2]|| PrimeOmega[#+2]==2)&&OddQ[Sqrt[ 1+8(#+2)]]&] (* Harvey P. Dale, Apr 12 2021 *)

A109799 Primes p such that 2^p - 1 is a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 127
Offset: 1

Views

Author

Jason Earls, Aug 15 2005

Keywords

Comments

For p in this sequence, 2^p - 1 is called a Mersenne-Chen prime.
Conjecture: 2^127 - 1 is the largest Mersenne-Chen prime.
Except for the initial term 2, this sequence is the intersection of A000043 and A000978 given by A107360. - Max Alekseyev, Oct 28 2008, Jan 28 2010

Examples

			a(5)=13 because 2^13 - 1 = 8191 is prime and 2^13 + 1 = 3*2731 is semiprime.

Crossrefs

Cf. A000043, A000978, A107360, A109611.

Programs

  • Mathematica
    Select[Prime[Range[40]],PrimeQ[2^#-1]&&PrimeOmega[2^#+1]<3&] (* James C. McMahon, Mar 30 2024 *)

A109807 Numbers n such that n^2 + 1 is a Chen prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 56, 66, 74, 94, 116, 120, 130, 134, 146, 160, 170, 176, 204, 230, 250, 256, 260, 284, 314, 326, 340, 350, 386, 406, 430, 440, 444, 464, 466, 470, 490, 496, 536, 556, 570, 584, 634, 646, 654, 680, 686, 700, 704, 714
Offset: 1

Views

Author

Jason Earls, Aug 16 2005

Keywords

Links

Crossrefs

Subsequence of A005574.
Cf. A005574, A109611.

Programs

  • Mathematica
    Select[Range[800],PrimeQ[#^2+1]&&PrimeOmega[#^2+3]<3&] (* Harvey P. Dale, May 07 2015 *)
  • PARI
    is(n)=isprime(n^2+1) && bigomega(n^2+3)<3 \\ Charles R Greathouse IV, Aug 19 2015

A109875 Chen primes p such that their p + 2 counterpart is a golden semiprime.

Original entry on oeis.org

13, 587, 1361, 15227, 118967, 337721, 383267, 512891, 1027331, 1780151, 2303681, 8200391, 9310517, 14666579, 25005089, 29105981, 34824971, 38895497, 40436909, 51819461, 63462977, 65427749, 65599199, 66043091, 75552479, 94671671
Offset: 1

Views

Author

Jason Earls, Aug 31 2005

Keywords

Comments

Conjecture: sequence is infinite.

Examples

			1361 is a term because it is prime and 1363 = 29*47 and abs(29*phi - 47) = 0.07701... < 1.

Links

Crossrefs

Cf. A109611, A108540.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p = 1; Do[p = NextPrime[p]; q = f[p]; If[q > 0 && PrimeQ[p*q - 2], AppendTo[seq, p*q - 2]], {1000}]; seq (* Amiram Eldar, Nov 29 2019 *)

Extensions

a(15)-a(26) from Donovan Johnson, Nov 17 2008

A109994 Chen primes p such that their p + 2 counterpart is a Sarrus number (pseudoprime to base 2).

Original entry on oeis.org

2699, 4679, 10259, 14489, 18719, 19949, 31607, 42797, 49139, 85487, 90749, 104651, 129887, 226799, 294269, 396269, 422657, 458987, 481571, 665279, 710531, 729059, 1082399, 1251947, 1302449, 1994687, 2035151, 2510567, 2811269, 3090089
Offset: 1

Views

Author

Jason Earls, Sep 01 2005

Keywords

Comments

Conjecture: sequence is infinite.

Links

Crossrefs

Cf. A109611, A001567.

Programs

  • Mathematica
    Select[Range[10^5], PrimeQ[#] && PowerMod[2, (# + 1), # + 2] == 1 && PrimeNu[# + 2] == 2 &] (* Amiram Eldar, Jun 28 2019 *)

A117242 Chen primes that are not twin primes.

Original entry on oeis.org

2, 23, 37, 47, 53, 67, 83, 89, 113, 127, 131, 157, 167, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 577, 587, 631, 647, 653, 677, 683, 701, 719, 743, 751, 761, 769, 787, 797, 839, 863, 877, 887, 911, 919, 937, 941, 947, 953, 971, 977, 983, 991
Offset: 1

Views

Author

Jani Melik, Apr 22 2006

Keywords

Examples

			a(1) = 2, 2 is a Chen prime but is not in a twin prime pair.
a(2) = 23 is a Chen prime, but is not in a twin prime pair.

Links

Crossrefs

Cf. A001097, A109611.

Programs

  • Maple
    ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end:
ts_chen_non_twin_primes:=proc(n) local i, ans; ans:=[ ]: for i from 1 to n do if (ischenprime(i) = 'true') and (isprime(i+2) = 'false' and isprime(i-2) = 'false') then ans:=[op(ans), i]: fi od; RETURN(ans) end:
ts_chen_non_twin_primes(1000);
  • Mathematica
    Lim=PrimePi[1000];Select[Select[Prime[Range[Lim]],PrimeOmega[#+2]<3&],!MemberQ[Select[ Prime[ Range[Lim]], PrimeQ[ # - 2]||PrimeQ[#+2]&] ,#]&] (* James C. McMahon, Sep 27 2024 *)
  • PARI
    is(n)=isprime(n)&&bigomega(n+2)==2&&!isprime(n-2) \\ Charles R Greathouse IV, May 04 2013
Previous Showing 21-30 of 71 results. Next

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